Suppose small aircraft arrive at a certain airport according to a Poisson process with a rate of 8 per hour. The number of arrivals during a time period of t hours is a Poisson distribution with parameter [tex]\mu = 8t[/tex].

(Round your answers to three decimal places.)

(a) What is the probability that exactly 8 small aircraft arrive during a 1-hour period?

(b) What is the probability that at least 8 small aircraft arrive during a 1-hour period?

(c) What is the probability that at least 13 small aircraft arrive during a 1-hour period?

The question pertains to calculating probabilities based on the Poisson distribution. The formulas given are used to calculate the exact and at least probabilities for the number of aircraft arrivals per hour.

Explanation:

The question involves calculations based on the Poisson distribution, which is a mathematical concept used in probability theory. It specifically deals with the probability of a given number of events (in this case, aircraft arrivals) happening in a fixed interval of time.

(a) To find the probability that exactly 8 aircrafts will arrive in 1 hour, we use the formula of Poisson distribution: P(X=k) = (λ^k * e^-λ) / k!. Here, λ (lambda) is the average rate, which is 8. So, P(X=8) =(8^8 * e^-8) / 8!.

For the probability that at least 8 small aircraft arrive in a 1-hour period, we need to sum the probabilities for 8, 9, 10 and so on till ∞. This can be calculated as: P(X >=8) = 1 - P(X < 8). We subtract the sum of probabilities of getting less than 8 aircraft from 1.

Similarly, for the probability that atleast 13 small aircrafts arrive in 1 hour, we will calculate: P(X >=13) = 1 - P(X < 13). Note: e here is Euler's number, a constant approximately equal to 2.71828. You would use a calculator (or software) capable of handling complex calculations to find these probabilities to three decimal places.

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